[Paper Review] A Computational Investigation on the Strength of Dantzig-Wolfe Reformulations

Bastubbe, Michael, Lübbecke, Marco E.+1 more|arXiv (Cornell University)|Dec 11, 2016

Computational Geometry and Mesh Generation39 references7 citations

TL;DR

This paper proposes a computationally efficient linear estimator for the distinct elements problem using discrete polynomial approximation and interpolation, achieving optimal sample complexity within O(log log k) factors. The key contribution is a sharp bound on the minimum singular values of real rectangular Vandermonde matrices, enabling near-optimal estimation of the number of distinct colors in an urn with n samples, valid for both sampling with and without replacement when sample size is small relative to urn size.

ABSTRACT

In Dantzig-Wolfe reformulation of an integer program one convexifies a subset of the constraints, leading to potentially stronger dual bounds from the respective linear programming relaxation. As the subset can be chosen arbitrarily, this includes the trivial cases of convexifying no and all constraints, resulting in a weakest and strongest reformulation, respectively. Our computational study aims at better understanding of what happens in between these extremes. For a collection of integer programs with few constraints we compute, optimally solve, and evaluate the relaxations of all possible (exponentially many) Dantzig-Wolfe reformulations (with mild extensions to larger models from the MIPLIBs). We observe that only a tiny number of different dual bounds actually occur and that only a few inclusion-wise minimal representatives exist for each. This aligns with considerably different impacts of individual constraints on the strengthening the relaxation, some of which have almost no influence. In contrast, types of constraints that are convexified in textbook reformulations have a larger effect. We relate our experiments to what could be called a hierarchy of Dantzig-Wolfe reformulations.

Motivation & Objective

• To determine the fundamental sample complexity for estimating the number of distinct colors in an urn using n samples.
• To develop a computationally efficient estimator with provable additive error guarantees for the distinct elements problem.
• To establish tight bounds on the minimum singular values of real rectangular Vandermonde matrices, which are central to the analysis.
• To unify and compare sampling models—multinomial, hypergeometric, Bernoulli, and Poisson—under a common framework.

Proposed method

• Uses discrete polynomial approximation and interpolation to construct linear estimators for the distinct elements problem.
• Applies ℓ2-approximation and Lagrange interpolation to derive estimators with optimal error bounds.
• Derives a sharp lower bound on the minimum singular value of real rectangular Vandermonde matrices, which is crucial for estimator stability.
• Employs correlation decay analysis between fingerprint statistics under the Poisson sampling model to control estimation error.
• Uses uniform asymptotic expansions of Stirling numbers of the first kind to analyze coefficient growth in interpolation polynomials.
• Establishes sample complexity bounds via minimax risk analysis under quadratic and zero-one loss, linking to minimax mean squared error.

Experimental results

Research questions

• RQ1 What is the optimal sample complexity for estimating the number of distinct elements in a population of size k with additive error ∆?
• RQ2 How can discrete polynomial approximation and interpolation be used to construct estimators with provable error guarantees in the distinct elements problem?
• RQ3 What is the sharp lower bound on the minimum singular value of a real rectangular Vandermonde matrix, and how does it impact estimator performance?
• RQ4 How do different sampling models (with/without replacement, Poisson, multinomial) relate in terms of fundamental sample complexity?
• RQ5 What is the minimax mean squared error for the distinct elements problem across different regimes of ∆ and k?

Key findings

• The sample complexity for estimating distinct elements within additive error ∆ is Θ(k log k / ∆²) when ∆ ≤ √k (log k)^{-δ}, achieved via interpolation-based estimators.
• For ∆ ∈ [k^{0.5+δ}, ck], the sample complexity is Θ(k / log k), matching the general support size estimation problem.
• The estimator via interpolation achieves ∆ = √k with n = O(k log log k) samples, and requires strictly superlinear sample size for ∆ ≤ k^{0.5−o(1)}.
• A sharp lower bound on the minimum singular value of real rectangular Vandermonde matrices is established, which is of independent mathematical interest.
• The minimax mean squared error is characterized as exp(−Θ(min{n log k / k, log k})) for n ≤ k log k, with phase transitions at n = k / log k and n = k.
• The paper shows that the multinomial and Poisson sampling models are asymptotically equivalent in terms of sample complexity, with explicit bounds on the transformation between them.